# Thread: Groups exercise questions

1. ## Groups exercise questions

I was wondering if someone could help with following exercise (image attached):

2. To show something is a subgroup, you need to show it contains the identity, $\displaystyle I_2$. It is closed under multiplication and inverses.

You are capable of doing this, it is very good practice.

1) if b and c are 0 you are just talking about diagonal matrices, but also note since they are in GL(2,R) neither of the diagonal entries can be 0 or else they will not be invertible since the determinant is the product of the diagonal entries.
Certainly the identity is in there. Write out multiplication of diagonal matrices, is it diagonal? What about inverses? is the inverse of a diagonal matrix diagonal the answer should be clear from what you got when you multiplied two arbitrary diagonal matrices (it is important here that neither entry is 0). See why?

3) is clearly not since it is not invertible if the entries must be integers. consider the one with both diagonal entries 2. It's inverse is the one with both diagonal entries 1/2, but that is not an integer, so it is not in the set, so its not closed under inverse.

4) Hint this is called the special linear GROUP. The chances are it is a group. When checking it recall $\displaystyle det(AB)=det(A)det(B)=1*1=1$
and $\displaystyle 1=det(I_2)=det(AA^{-1})=det(A)det(A^{-1}) \Rightarrow det(A^{-1})=\frac{1}{det(A)}=\frac{1}{1}=1$

5) no what is the determinant of a matrix with first row all 0s? Are empty sets groups?

2) I leave to you, it will make you stronger.